Additive crystal-field model

The one-electron crystal field Hamiltonian does not take into account electron correlation effects. For some systems, it has been useful to augment the crystal field Hamiltonian with additional terms representing the two-electron, correlated crystal field. The additional terms most commonly used (see, for example, Peijzel et al., 2005b Wegh et al., 2003) are from the simplified delta-function correlation crystal field model first proposed by Judd (1978) that assumes electron interaction takes place only when two electrons are located at the same position (hence the name delta-function ). This simplified model, developed by Lo and Reid (1993), adds additional terms, given as,... 

We have seen that the crystal-field model provides a basis for explaining many features of transition-metal complexes. In fact, it can be used to explain many observations in addition to those we have discussed. Many lines of evidence show, however, that the bonding between transition-metal ions and ligands must have some covalent character. Molecular-orbital theory (Sections 9.7 and 9.8) can also be used to describe the bonding in complexes, although the application of molecular-orbital theory to coordination compounds is beyond the scope of our discussion. The crystal-field model, although not entirely accurate in all details, provides an adequate and useful first description of the electronic structure of complexes. 

In addition to modifications of the atomic parameters, there are medium-related effects which must be taken into account explicitly. The broken spherical symmetry that normally results when an isolated free gaseous ion is placed in a ligand field gives rise to a splitting of the free-ion level into a maximum of 2J +1) components. A single-particle crystal field model has had remarkable success for lanthanides and somewhat qualified, but nevertheless satisfactory, success for the trivalent actinides in providing an interpretation of the data [3, 6, 24]. The additional splitting induced by the crystal field can be described by the expression ... 

Mallinson et al. (1988) have performed an analysis of a set of static theoretical structure factors based on a wave function of the octahedral, high-spin hexa-aquairon(II) ion by Newton and coworkers (Jafri et al. 1980, Logan et al. 1984). To simulate the crystal field, the occupancy of the orbitals was modified to represent a low-spin complex with preferential occupancy of the t2g orbitals, rather than the more even distribution found in the high-spin complex. The complex ion (Fig. 10.14) was centered at the corners of a cubic unit cell with a = 10.000 A and space group Pm3. Refinement of the 1375 static structure factors (sin 8/X < 1.2 A 1) gave an agreement factor of R = 4.35% for the spherical-atom model with variable positional parameters (Table 10.12). Addition of three anharmonic thermal... 

The geometric model adopted here is one in which the O2 molecule provides a C2v component to the crystal field potential a-round the Cu11 ion, and together with the three framework oxygens of the SII site makes an additional tetrahedral (T ) component. This model has been theoretically analyzed earlier (30) and the... 

A general model for electronic relaxation of the Gd3+ S = 7/2 ion in various complexes in solution was presented by Rast el al. [86]. Contrary to the usual assumption, the electronic relaxation in their model is not only due to the effects of the transient zero field splitting, but is also strongly influenced by the static crystal field effect which is modulated by the random Brownian rotation of the complex. Experimental peak-to-peak widths of three gadolinium complexes could be well interpreted as a function of temperature and frequency using three static and one transient crystal field parameters. Moreover, their interpretation of experimental data did not require the addition of any field independent contribution to the line width like the spin-rotation mechanism. 

The crystal-field parameters introduced in sect. 4.1 still contain all the structural information about the local environment. Therefore, a direct comparison of crystal-field parameters derived from different hosts, even with the same site symmetry, is not reasonable. In addition, the crystal-field parameters cannot be directly related to the distance and angle variations induced by the high-pressure application. Widely used models which extract the structural information from the crystal-field parameters are the angular-overlap (Jprgensen et al., 1963) and superposition model (Bradbury and Newman, 1967). In the case of f elements, the superposition model has been employed widely for the analysis of crystal-field parameters. 

The underlying assumption of the superposition model is that the one-electron crystal field is additive and can be regarded as a superposition of the contributions from individual ions... 

As a last remark, it should be mentioned that also a few evaluations of the crystal-field parameters of Pr3"1" in LaCH in the scope of the angular overlap model have been made. Urland et al. (1985) andUrland (1989) used the angular overlap model to calculate the crystal-field splittings of LaCHiPr3"1" under pressure. In addition, Gregorian et al. (1989) derived the parameters of both models and found that both approaches were capable to successfully describe the high-pressure results. 

It must be kept in mind, that S represents just a first-order approximation of the distribution function, and this under the additional premise of complete cylindrical symmetry only. It might be an acceptable measure when comparing cases for which a mean-field model applies. However, comparing the order parameters of liquid crystals with those of other partially ordered phases, such as stretched polymers or tribological samples can be misleading due to possibly different types of distribution functions. 

The introduction and implementation of heteronuclear-based multidimensional techniques have revolutionized the protein NMR field. Large proteins (> 100 residues) are now amenable to detailed NMR studies and structure determination. These techniques, however, necessarily require a scheme by which and isotopes can be incorporated into the protein to yield a uniformly labeled sample. Additional complications, such as extensive covalent post-translational modifications, can seriously limit the ability to efficiently and cost effectively express a protein in isotope enriched media - the c-type cytochromes are an example of such a limitation. In the absence of an effective labeling protocol, one must therefore rely on more traditional proton homonuclear NMR methods. These include two-dimensional (1) and, more recently, three-dimensional H experiments (2,3). Cytochrome c has become a paradigm for protein folding and electron transfer studies because of its stability, solubility and ease of preparation. As a result, several high-resolution X-ray crystal structure models for c-type cytochromes, in both redox states, have emerged. Although only subtle structural differences between redox states have been observed in these... 

Among the early successes of crystal field theory was its ability to account for magnetic and spectral properties of complexes. In addition, it provided a basis for understanding and predicting a number of their structural and thermodynamic properties. Several such properties are described in this section from the crystal field point of view. Certainly other bonding models, such as molecular orbital theory, can also be used to interpret these observations. Even when they are, however, concepts from crystal field theory, such as crystal (or ligand) field stabilization energy, are often invoked within the discussion. 

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